Paper 4, Section I, D

Numbers and Sets | Part IA, 2017

(a) Give the definitions of relation and equivalence relation on a set SS.

(b) Let Σ\Sigma be the set of ordered pairs (A,f)(A, f) where AA is a non-empty subset of R\mathbb{R} and f:ARf: A \rightarrow \mathbb{R}. Let R\mathcal{R} be the relation on Σ\Sigma defined by requiring (A,f)R(B,g)(A, f) \mathcal{R}(B, g) if the following two conditions hold:

(i) (A\B)(B\A)(A \backslash B) \cup(B \backslash A) is finite and

(ii) there is a finite set FABF \subset A \cap B such that f(x)=g(x)f(x)=g(x) for all xAB\Fx \in A \cap B \backslash F.

Show that R\mathcal{R} is an equivalence relation on Σ\Sigma.

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